Integration of cot(x)/sin(x)dx: A Comprehensive Guide

Understanding how to integrate trigonometric functions like cot(x)/sin(x) is crucial in various mathematical applications and problem-solving scenarios. This guide aims to provide a detailed explanation of how to solve this specific integral, breaking down the process step-by-step. Whether you're a student, a professional, or just curious about mathematics, this article is designed to help you tackle this integral confidently.

Introduction to the Integral∫ cot(x)/sin(x) dx

The integral of cot(x)/sin(x) can be approached using various methods from calculus, particularly those related to trigonometric functions and their properties. The integral is given by:

∫ cot(x)/sin(x) dx ∫ cos(x) / (sin2(x)) dx

Solving the Integral∫ cot(x)/sin(x) dx

To solve the integral, we can use a substitution method. Let's start by simplifying the given integral:

∫ cos(x) / (sin2(x)) dx

Substitution: Let sin(x) t

Using the substitution sin(x) t, we get the differential:

cos(x) dx dt

Now, substitute sin(x) t and cos(x) dx dt into the integral:

∫ dt / (t2)

This simplifies to:

∫ 1/t2 dt

Integrating 1/t2

The integral of 1/t2 is straightforward:

∫ 1/t2 dt -1/t C

Substituting back

Substitute back t sin(x) into the result:

-1/t C -1/sin(x) C

Final Result

Therefore, the integral of cot(x)/sin(x) dx is:

∫ cot(x)/sin(x) dx -1/sin(x) C

Alternatively, this can be written as:

∫ cot(x)/sin(x) dx -cosec(x) C

Another Approach: Using Trigonometric Identities

Alternatively, you can solve the integral using trigonometric identities directly:

∫ cot(x)/sin(x) dx ∫ (cos(x)/sin(x)) / sin(x) dx ∫ (cos(x)/sin2(x)) dx

Right-hand side: ∫ -cosec(x) cot(x) dx

Note that:

cosec(x) 1/sin(x) and cot(x) cos(x)/sin(x)

Therefore:

∫ (cos(x)/sin2(x)) dx ∫ -cosec(x) cot(x) dx

The integral of -cosec(x) cot(x) with respect to x is:

∫ -cosec(x) cot(x) dx -cosec(x) C

Conclusion

In summary, the integral of cot(x)/sin(x) dx is a bit tricky and requires either a substitution or direct use of trigonometric identities. Both methods yield the same result: -cosec(x) C. Understanding these techniques is essential for mastering integral calculus and solving a wide range of mathematical problems.

Frequently Asked Questions (FAQ)

1. What is the integral of cot(x)/sin(x) dx?

The integral of cot(x)/sin(x) dx is -cosec(x) C.

2. Can this integral be solved using substitution?

Yes, by letting sin(x) t and using cos(x) dx dt, the integral becomes ∫ 1/t2 dt, which can be solved as -1/t C -1/sin(x) C.

3. How can this integral be approached using identities?

Using the identities cosec(x) 1/sin(x) and cot(x) cos(x)/sin(x), the integral can be directly solved as ∫ -cosec(x) cot(x) dx, resulting in -cosec(x) C.